Question

Will someone please show me step by step how to do this problem? Thanks!
The automobile assembly plant you manage has a Cobb-Douglas production function given by
P = 20x^0.7y^0.3
where P is the number of automobiles it produces per year, x is the number of employees, and y is the daily operating budget (in dollars). Assume that you maintain a constant work force of 130 workers and wish to increase production in order to meet a demand that is increasing by 50 automobiles per year. The current demand is 1000 automobiles per year. How fast should your daily operating budget be increasing? HINT [See Example 4.] (Round your answer to the nearest cent.)

Answer 1

ok, what have you tried so far?

Answer 2

I'm not positive where to begin. So far I have:
50 = 20x^0.7y^0.3 * dx/dt...

Answer 3

hang on . I will be right back...

Answer 4

Okay thank you so much for helping!

Answer 5

ok,
P = 20x^.7y^0.3 but we are told x is constant...agreed?

Answer 6

The problem indicates that x = 130 constant
50 automobiles per year increase would be dP/dt
they are asking for how fast the operating budget increase which is dy/dt
Agree?

Answer 7

You there?

Answer 8

Yes sorry

Answer 9

no problem...do you agree with my assessment?

Answer 10

Yes so we are looking dy/dt?

Answer 11

Exactly...so from the equation I need to get dP/dt and dy/dt.
What rule in calculus do I use to get that from the original expression?

Answer 12

the chain rule?

Answer 13

Excellent ...so go ahead and use the chain rule and remember x is constant

Answer 14

how is it coming?

Answer 15

I think I'm confusing myself with this problem and how to set it up

Answer 16

ok, can you take the derivative of the equation using the chain rule remembering x is constant

Answer 17

do I take the derivative from P = 20x^0.7y^0.3

Answer 18

yes ...use the chain rule so that you derive dP/dt and dy/dt terms and remember x is constant ...go ahead

Answer 19

So (14y^3/10)/(x^3/10)

Answer 20

Not quite...

Answer 21

P = 20x^0.7y^0.3
so according to the chain rule
dP/dt = 20x^.7 * .3y^-.7 dy/dt
Have a close look and lets make sure you understand this

Answer 22

Okay so then do I plug in for the x and y?

Answer 23

So for Y:
P = P = 20x^0.7y^0.3
1000 = 20(130)^.7*y^.3 ?

Answer 24

exactly...

Answer 25

And then x is 130?

Answer 26

yes...
so now you have dp/dt = 50, x = 130, y = whatever you solve for, the only thing left is to get dy/dt which is what the problem wants

Answer 27

dP/dt = 20x^.7 * .3y^-.7 dy/dt this is what you use to get dy/dt

Answer 28

do you feel you understand the problem now?

Answer 29

Yes thank you so much!

Answer 30

Fantastic...have a good night!

Answer 31

Good night!